The Strain Energy Density of Compressible, Rubber-Like Shells of Revolution

1996 ◽  
Vol 63 (2) ◽  
pp. 419-423 ◽  
Author(s):  
R. F. Yu¨kseler
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Density Functions ◽  
Two Dimensional ◽  
Circular Plates ◽  
Shells Of Revolution ◽  
Large Rotations ◽  
Shear Strains ◽  
Transverse Shear Strains

An approach for the derivation of two-dimensional strain energy density functions of compressible, rubber-like shells of revolution which undergo axisymmetric, arbitrarily large rotations and strains including transverse normal and transverse shear strains is proposed. The proposed method is applied to polyurethane rubber, circular plates and numerical results for the flexural buckling of polyurethane circular plates are obtained for an illustrative comparison with a method presented previously.

Constitutive Equations for Hyperelastic Materials Based on the Upper Triangular Decomposition of the Deformation Gradient

2018 ◽  
Vol 24 (6) ◽  
pp. 1785-1799 ◽  
Author(s):  
Y. Q. Li ◽  
X.-L. Gao
Keyword(s):  
Energy Density ◽  
Constitutive Equations ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Deformation Gradient ◽  
Density Functions ◽  
Triangular Decomposition ◽  
Deformation Gradient Tensor ◽  
Hyperelastic Materials ◽  
Distortion Tensor

The upper triangular decomposition has recently been proposed to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular tensor called the distortion tensor, whose six components can be directly related to pure stretch and simple shear deformations, which are physically measurable. In the current paper, constitutive equations for hyperelastic materials are derived using strain energy density functions in terms of the distortion tensor, which satisfy the principle of material frame indifference and the first and second laws of thermodynamics. Being expressed directly as derivatives of the strain energy density function with respect to the components of the distortion tensor, the Cauchy stress components have simpler expressions than those based on the invariants of the right Cauchy-Green deformation tensor. To illustrate the new constitutive equations, strain energy density functions in terms of the distortion tensor are provided for unconstrained and incompressible isotropic materials, incompressible transversely isotropic composite materials, and incompressible orthotropic composite materials with two families of fibers. For each type of material, example problems are solved using the newly proposed constitutive equations and strain energy density functions, both in terms of the distortion tensor. The solutions of these problems are found to be the same as those obtained by applying the polar decomposition-based invariants approach, thereby validating and supporting the newly developed, alternative method based on the upper triangular decomposition of the deformation gradient tensor.

Peridynamic Modeling of Hyperelastic Membrane Deformation

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
D. J. Bang ◽  
E. Madenci
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Equations Of Motion ◽  
Uniaxial Loading ◽  
Density Functions ◽  
Loading Conditions ◽  
Element Analysis ◽  
Material Parameters ◽  
Loading Case

This study concerns the development of peridynamic (PD) strain energy density functions for a Neo-Hookean type membrane under equibiaxial, planar, and uniaxial loading conditions. The material parameters for each loading case are determined by equating the PD strain energy density to that of the classical continuum mechanics. The PD equations of motion are derived based on the Neo-Hookean model under the assumption of incompressibility. Numerical results concern the deformation of a membrane with a defect in the form of a hole, a crack, and a rigid inclusion under equibiaxial, planar, and uniaxial loading conditions. The PD predictions are verified by comparison with those of finite element analysis.

scholarly journals A generative model of hyperelastic strain energy density functions for multiple tissue brain deformation

2020 ◽  
Author(s):  
Alejandro Granados ◽  
Fernando Perez-Garcia ◽  
Martin Schweiger ◽  
Vejay Vakharia ◽  
Sjoerd B. Vos ◽  
...  
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Generative Model ◽  
Cavity Surface ◽  
Density Functions ◽  
Biomechanical Models ◽  
Mechanical Characterisation ◽  
Brain Deformation ◽  
Clinical Cases

Abstract Purpose Estimation of brain deformation is crucial during neurosurgery. Whilst mechanical characterisation captures stress–strain relationships of tissue, biomechanical models are limited by experimental conditions. This results in variability reported in the literature. The aim of this work was to demonstrate a generative model of strain energy density functions can estimate the elastic properties of tissue using observed brain deformation. Methods For the generative model a Gaussian Process regression learns elastic potentials from 73 manuscripts. We evaluate the use of neo-Hookean, Mooney–Rivlin and 1-term Ogden meta-models to guarantee stability. Single and multiple tissue experiments validate the ability of our generative model to estimate tissue properties on a synthetic brain model and in eight temporal lobe resection cases where deformation is observed between pre- and post-operative images. Results Estimated parameters on a synthetic model are close to the known reference with a root-mean-square error (RMSE) of 0.1 mm and 0.2 mm between surface nodes for single and multiple tissue experiments. In clinical cases, we were able to recover brain deformation from pre- to post-operative images reducing RMSE of differences from 1.37 to 1.08 mm on the ventricle surface and from 5.89 to 4.84 mm on the resection cavity surface. Conclusion Our generative model can capture uncertainties related to mechanical characterisation of tissue. When fitting samples from elastography and linear studies, all meta-models performed similarly. The Ogden meta-model performed the best on hyperelastic studies. We were able to predict elastic parameters in a reference model on a synthetic phantom. However, deformation observed in clinical cases is only partly explained using our generative model.

A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part II: Applications

1996 ◽  
Vol 63 (4) ◽  
pp. 869-876 ◽  
Author(s):  
Jiun-Shyan Chen ◽  
Cheng-Tang Wu ◽  
Chunhui Pan
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Projection Method ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Finite Elasticity ◽  
Element Formulation ◽  
Density Functions ◽  
Hyperelastic Materials ◽  
Pressure Projection

In the first part of this paper a pressure projection method was presented for the nonlinear analysis of structures made of nearly incompressible hyperelastic materials. The main focus of the second part of the paper is to demonstrate the performance of the present method and to address some of the issues related to the analysis of engineering elastomers including the proper selection of strain energy density functions. The numerical procedures and the implementation to nonlinear finite element programs are presented. Mooney-Rivlin, Cubic, and Modified Cubic strain energy density functions are used in the numerical examples. Several classical finite elasticity problems as well as some practical engineering elastomer problems are analyzed. The need to account for the slight compressibility of rubber (finite bulk modulus) in the finite element formulation is demonstrated in the study of apparent Young’s modulus of bonded thin rubber units. The combined shear-bending deformation that commonly exists in rubber mounting systems is also analyzed and discussed.

Equilibrium Shapes of Small Strained Islands

1995 ◽  
Vol 399 ◽  
Author(s):  
Brian J. Spencer ◽  
J. Tersoff
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Chemical Potential ◽  
Equilibrium Shape ◽  
Two Dimensional ◽  
Asymptotic Results ◽  
Island Size ◽  
Equilibrium Shapes ◽  
Equilibrium Morphology

ABSTRACTWe calculate the equilibrium morphology of a strained layer, for the case where it wets the substrate (Stranski-Krastonow growth). Assuming isotropic surface energy and equal elastic constants in the film and substrate, we are able to calculate two-dimensional equilibrium shapes as a function of the island size and spacing. We present asymptotic results for the equilibrium shape of a thin island where the island height is much smaller than the island width. We also present numerical results of the full equations to describe the island shape when the islands are widely separated. From these solutions we are able to determine the chemical potential of the island as a function of island volume and the strain energy density along the surface of the island for small to medium-sized islands.

scholarly journals The anelastic Ericksen problem: universal eigenstrains and deformations in compressible isotropic elastic solids

2016 ◽  
Vol 472 (2196) ◽  
pp. 20160690 ◽  
Author(s):  
Arash Yavari ◽  
Alain Goriely
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Natural Generalization ◽  
Density Functions ◽  
Elastic Solids ◽  
Connected Body ◽  
Simply Connected ◽  
Deformation Gradients ◽  
Constant Deformation

The elastic Ericksen problem consists of finding deformations in isotropic hyperelastic solids that can be maintained for arbitrary strain-energy density functions. In the compressible case, Ericksen showed that only homogeneous deformations are possible. Here, we solve the anelastic version of the same problem, that is, we determine both the deformations and the eigenstrains such that a solution to the anelastic problem exists for arbitrary strain-energy density functions. Anelasticity is described by finite eigenstrains. In a nonlinear solid, these eigenstrains can be modelled by a Riemannian material manifold whose metric depends on their distribution. In this framework, we show that the natural generalization of the concept of homogeneous deformations is the notion of covariantly homogeneous deformations —deformations with covariantly constant deformation gradients. We prove that these deformations are the only universal deformations and that they put severe restrictions on possible universal eigenstrains . We show that, in a simply-connected body, for any distribution of universal eigenstrains the material manifold is a symmetric Riemannian manifold and that in dimensions 2 and 3 the universal eigenstrains are zero-stress.

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